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Caustics of Weakly Lagrangian Distributions

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Abstract

We study semiclassical sequences of distributions \(u_h\) associated with a Lagrangian submanifold of phase space \(\mathcal {L}\subset T^*X\). If \(u_h\) is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on \(\mathcal {L},\) then the asymptotics of \(u_h\) are well understood by work of Arnol’d, provided \(\mathcal {L}\) projects to X with a stable simple Lagrangian singularity. We establish sup-norm estimates on \(u_h\) under much more general hypotheses on the rate at which it is concentrating on \(\mathcal {L}\) (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians.

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Authors and Affiliations

  1. Northwestern University, Evanston, USA

    Seán Gomes & Jared Wunsch

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  1. Seán Gomes
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  2. Jared Wunsch
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Communicated by Stéphane Nonnenmacher.

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The authors are grateful to Steve Zelditch for helpful discussions and to Ilya Khayutin for explaining the number-theoretic literature on lattice point counting in shrinking spherical caps (Sect. 2). Stéphane Nonnenmacher as well as two anonymous referees made many helpful suggestions on the exposition; one of the latter pointed out an error in the inductive step proving the main theorem. JW gratefully acknowledges partial support from Simons Foundation Grant 631302 and from NSF Grant DMS–1600023.

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Gomes, S., Wunsch, J. Caustics of Weakly Lagrangian Distributions. Ann. Henri Poincaré 23, 1205–1237 (2022). https://doi.org/10.1007/s00023-021-01110-8

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